The field of the invention is systems and methods for medical imaging. More particularly, the invention relates to systems and methods for medical image reconstruction.
Magnetic resonance imaging (“MRI”) uses the nuclear magnetic resonance (“NMR”) phenomenon to produce images. When a substance such as human tissue is subjected to a uniform magnetic field (polarizing field B0), the individual magnetic moments of the nuclei in the tissue attempt to align with this polarizing field, but precess about it in random order at their characteristic Larmor frequency. If the substance, or tissue, is subjected to a magnetic field (excitation field B1) that is in the x-y plane and that is near the Larmor frequency, the net aligned moment, Mz, may be rotated, or “tipped,” into the x-y plane to produce a net transverse magnetic moment Mxy. A signal is emitted by the excited nuclei or “spins,” after the excitation signal B1 is terminated, and this signal may be received and processed to form an image.
When utilizing these “MR” signals to produce images, magnetic field gradients (Gx, Gy, and Gz) are employed. Typically, the region to be imaged is scanned by a sequence of measurement cycles in which these gradients vary according to the particular localization method being used. The resulting set of received MR signals are digitized and processed to reconstruct the image using one of many well known reconstruction techniques.
The measurement cycle used to acquire each MR signal is performed under the direction of a pulse sequence produced by a pulse sequencer. Clinically available MRI systems store a library of such pulse sequences that can be prescribed to meet the needs of many different clinical applications. Research MRI systems include a library of clinically-proven pulse sequences and they also enable the development of new pulse sequences.
In clinical applications of MRI, it is often beneficial to image the same region of interest under multiple contrast settings because this makes it possible to emphasize different tissue types. The fact that different pathologies exhibit different relaxation times makes multi-contrast scans very valuable for diagnostics.
Recently, a new mathematical framework for image reconstruction termed “compressed sensing” (“CS”) was formulated. According to compressed sensing theory, only a small set of linear projections of a compressible image are required to reconstruct an accurate image. The theory of CS is described, for example, by E. Candès, et al., in “Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information,” IEEE Transactions on Information Theory, 2006; 52:489-509; and by D. Donoho in “Compressed Sensing,” IEEE Transactions on Information Theory, 2006; 52:1289-1306; and is disclosed, for example, in U.S. Pat. No. 7,646,924. Given a set of underdetermined linear equations:y=φx  (1);
where x∈M, y∈K, and Φ∈K×M with K<M, compressed sensing theory aims to find the sparsest solution in an orthonormal basis, Ψ, by solving the following constrained optimization problem:
                              x          ^                =                                                            arg                ⁢                                                                  ⁢                min                            x                        ⁢                                                                                                Ψ                    T                                    ⁢                  x                                                            1                        ⁢                                                  ⁢            such            ⁢                                                  ⁢            that            ⁢                                                  ⁢            y                    =                      Φ            ⁢                                                  ⁢                          x              .                                                          (        2        )            
Magnetic resonance imaging is one of the areas where compressed sensing has received abundant attention. This is because the data collected during an MRI scan are in Fourier space, or “k-space”; therefore, reconstructing images from this data involves solving an inverse problem. Because the duration of an MRI scan is related to the number of data points that are sampled in k-space, it is of interest to obtain compressive samples and speed up data acquisition. Making use of CS theory to this end was first proposed by M. Lustig, et al., in “Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging,” Magn. Reson. Med., 2007; 58(6): 1182-95, in which the inversion problem was formulized as:
                                          x            ^                    =                                                                                          arg                    ⁢                                                                                  ⁢                    min                                    x                                ⁢                                                                                                                        Ψ                        T                                            ⁢                      x                                                                            1                                            +                                                β                  ·                                      TV                    ⁡                                          (                      x                      )                                                                      ⁢                                                                  ⁢                such                ⁢                                                                  ⁢                that                ⁢                                                                  ⁢                                                                                                y                      -                                                                        F                          Ω                                                ⁢                        x                                                                                                  2                                                      <            ɛ                          ;                            (        3        )            
where y is allowed to be complex valued, Ψ is a wavelet basis, TV ( . . . ) indicates the total variation operation, β is a parameter that trades wavelet sparsity with finite differences sparsity, FΩ is an undersampled Fourier transform operator containing only the frequencies ω∈Ω, and ε is a threshold parameter that is tuned for each reconstruction task.
In light of the foregoing, it would be desirable to provide an image reconstruction method that can produce accurate images from few data samples, such as compressed sensing, but can do so while simultaneously reconstructing multiple images from a plurality of different data sets acquired with different contrast characteristics. Currently available image reconstruction methods that employ compressed sensing do not utilize information sharing among multiple image data sets acquired with differing contrast or other imaging characteristics.